Spaces with Ricci curvature bounded below

微分几何是数学的一个分支，研究光滑空间的性质，称为流形，它在数学，物理学和其他科学学科中有着深远的应用。一般来说，测量流形形状的那些量称为曲率。对一种特殊的曲率--里奇曲率--的研究是微分几何的中心课题之一，与广义相对论有着重要的联系。本项目旨在研究Ricci曲率不能太低的空间的几何和拓扑，即，这里的曲率有一个下限 Ricci曲率下界的概念也可以扩展到奇异空间（例如，具有锥尖的空间），其可以不具有歧管结构。对这些奇异空间的研究可以进一步加深对光滑流形上Ricci曲率的理解。该项目的更广泛的影响包括通过研讨会和会议演讲提高人们对奇异空间重要性的认识，以易于理解的方式将几何学中的现代概念和思想带给学生，并指导这些领域的本科生。在一个方向上，该项目将研究具有非负Ricci曲率的完备和非紧流形的基本群。具体来说，这将研究基本群的结构（例如，有限生成和虚幂零性）与等变渐近几何之间的关系。该项目还旨在了解Ricci极限空间的几何，特别是具有大Hausdorff维数的奇异集的几何。最后，该项目将探索Ricci曲率下有界的闭流形的基本群，包括它们在Gromov-Hausdorff拓扑下的稳定性和对群结构的一致控制。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估而被认为值得支持。

Differential geometry is a branch of mathematics that studies the properties of smooth spaces, known as manifolds, and it has profound applications throughout mathematics, physics, and other scientific disciplines. In general, those quantities that measure the shape of the manifold are called curvature. The study of one particular kind of curvature -- Ricci curvature -- is one of central topics in differential geometry and has significant connections to general relativity. This project seeks to study the geometry and topology of spaces whose Ricci curvature can't be too low, i.e., where this curvature has a lower bound. The notion of Ricci curvature lower bounds can also be extended to singular spaces (e.g., spaces with cone tips) which may not have manifold structures. Investigating these singular spaces can in turn advance the understanding of Ricci curvature on smooth manifolds. The broader impacts of this project include raising awareness of the importance of singular spaces through seminar and conference presentations, bringing modern concepts and ideas in geometry to students in an accessible way, as well as mentoring undergraduates in these areas.In one direction, the project will study the fundamental groups of complete and non-compact manifolds with nonnegative Ricci curvature. Specifically, this will investigate the relation between the structure of fundamental groups (for example, finite generation and virtual nilpotency) and the equivariant asymptotic geometry. The project will also aim to understand the geometry of Ricci limit spaces, especially that of singular sets with large Hausdorff dimension. Lastly, the project will explore the fundamental groups of closed manifolds with Ricci curvature bounded below, including their stability under the Gromov-Hausdorff topology and uniform control on the group structure.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.